Subgroups — #12445

Order 256.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2224	C₂	1	2	yes	maximal
2	#1713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈	C₂	1	1	yes	maximal, characteristic
4	#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	#1364	C₂	1	2	yes	maximal, complemented
6	#500	—	2	1	no	—
7	#500	C₂²	1	2	yes	—
8	#494	—	2	1	no	—
9	C₄×D₈	C₂²	1	1	yes	characteristic
10	C₈×D₄	—	2	1	no	—
11	#366	—	2	2	no	—
12	C₄⋊C₁₆	C₂²	1	2	yes	—
13	#357	—	2	2	no	—
14	#335	C₂²	1	1	yes	characteristic
15	D₄⋊C₈	—	2	1	no	—
16	C₈²	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	—	4	1	no	—
18	C₂×Q₁₆	—	4	1	no	—
19	C₂×D₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	—	4	1	no	—
21	C₄⋊Q₈	—	2	2	no	—
22	C₄×D₄	—	2	1	no	—
23	C₂×C₁₆	—	2	2	no	—
24	#107	—	2	2	no	—
25	#107	—	2	2	no	—
26	#107	—	4	2	no	—
27	#107	D₄	1	1	yes	characteristic
28	C₄⋊C₈	—	2	1	no	—
29	#102	—	2	1	no	—
30	C₂²⋊C₈	—	4	1	no	—
31	C₄×C₈	D₄	1	1	yes	characteristic
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₄×C₈	C₂³	1	1	yes	characteristic
34	C₂×Q₈	—	4	2	no	—
35	C₂×D₄	—	2	1	no	—
36	C₂²×C₄	—	4	1	no	—
37	Q₁₆	—	8	2	no	—
38	Q₁₆	—	8	2	no	—
39	D₈	—	2	1	no	—
40	D₈	Q₁₆	1	2	yes	complemented
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	8	2	no	—
44	C₂×C₈	D₈	1	1	yes	characteristic
45	C₂×C₈	D₈	1	1	yes	characteristic
46	C₂×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈	#41	1	1	yes	characteristic
48	C₄⋊C₄	—	2	1	no	—
49	C₄⋊C₄	—	4	2	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₂²⋊C₄	—	4	1	no	—
52	C₄²	C₂×D₄	1	1	yes	characteristic
53	C₁₆	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	Q₈	—	8	2	no	—
56	Q₈	—	8	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	8	1	no	—
61	C₂×C₄	—	8	1	no	—
62	C₂×C₄	—	8	2	no	—
63	C₂×C₄	#121	1	1	yes	characteristic
64	C₂×C₄	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₈	—	8	1	no	—
70	C₈	D₁₆	1	2	yes	—
71	C₈	C₂×Q₁₆	1	1	yes	characteristic
72	C₈	#135	1	1	yes	characteristic
73	C₂²	—	4	2	no	—
74	C₂²	—	8	1	no	—
75	C₂²	—	8	1	no	—
76	C₂²	#459	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	8	1	no	—
79	C₄	—	16	2	no	—
80	C₄	#467	1	1	yes	characteristic
81	C₄	#473	1	1	yes	characteristic
82	C₄	C₂×D₁₆	1	1	yes	characteristic
83	C₄	#509	1	1	yes	characteristic
84	C₂	—	8	2	no	—
85	C₂	#1669	1	1	yes	characteristic
86	C₂	#2206	1	1	yes	characteristic
87	C₂	#2214	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #1713	C₂	semidirect
4. #1370	C₂	semidirect
5. #1364	C₂	semidirect
40. D₈	Q₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#2224	2
#1713	2
C₈×D₈	2
#1370	2
#1364	2